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MATH 701 – Combinatorial Analysis (3-0-3)

This is a graduate course to develop understanding of discrete mathematics through study of combinatorial principles. Topics include: Graphs; Trees; Coloring of graphs and Ramsey theorem; Dilworth鈥檚 theorem and extremal set theory; Inclusion-exclusion principle; Recursions and generating functions; Partitions; (0, 1)-Matrices; Hadamard matrices; Reed-Muller codes; Codes and designs; Strongly regular graphs and partial geometries; Projective and combinatorial geometries; Polya theory of counting.

MATH 702 – Functional analysis (3-0-3)

Functional analysis is a branch of mathematical analysis dealing with the study of normed, Banach, and Hilbert spaces endowed with some kind of limit-related structure such as for instance an inner product, norm, topology, etc. and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, e.g., continuous, unitary, etc. operators between function spaces. This point of view turns out to be particularly useful for the study of differential and integral equations.

MATH 703 – Finance and Stochastic Calculus (3-0-3)

The course gives an up-to-date and modern overview of the main concepts in Mathematical Finance in continuous time. This includes mastering the basic tools of the subject such as Wiener processes, martingales, It么 calculus and Feynman-Kac formula. The tools will then lead to the study of important problems such as portfolio optimization, hedging and replication in continuous time, as well as the calculus of option prices.

MATH 704 – Matrix Computations (3-0-3)

Reliable and efficient algorithms for solution of linear systems, least squares problems, the eigenvalue problems and related problems. Perturbation analysis of problems as well as stability and sensitivity of algorithms and certain conditions of problems. Algorithms for structured matrices. Designing and programming reliable numerical algorithms using MATLAB leading to high performance matrix computations codes.

MATH 705 – Mechanics of interacting particles (3-0-3)

The course gives an up-to-date and modern overview of the main concepts in the mechanics of interacting particles. Using the Lagrangian/Hamiltonian formalisms, the course continues with an axiomatic approach to quantum mechanics. Path integrals and path integral quantization of Bosonic and Fermionic particles are also treated. An introduction to gauge theories, the Higgs field and the Bosonic string theory are also given

MATH 706 – Modern Statistical Prediction and Data Mining (3-0-3)

This course will train students in a variety of modern statistical and computational methods that enable researchers to learn from data and make sense of the vast amounts of data being generated in many fields. Although emphasizing applications in computer code, this course will also require mathematical proofs and derivations. Techniques taught in this course include regularization, kernel smoothing, model selection, model inference (bootstrap and EM algorithm), additive models, boosting and additive trees, neural networks, support vector machines, and ensemble learning.聽聽 Particular emphasis is placed on understanding the strengths and weaknesses between different methodologies used for extracting patterns and trends from large and complex data.

MATH 707 – Nonlinear Optimization (3-0-3)

course in real analysis in finite dimensional spaces. Optimization has applications in many fields, both in research and industry, and forms an integral part of parameter estimation and control, as well as machine learning. This course aims to give doctoral students an overview of the current state of the art in nonlinear optimization. We will discuss the theoretical underpinnings and convergence of various optimization approaches and offer practical guidelines for their use and implementation. Topics include unconstrained and constrained optimization, linear and quadratic programming, Lagrange and conic duality theory, interior-point algorithms and theory, Lagrangian relaxation, generalized programming, and semi-definite programming. Algorithmic methods used in the class include steepest descent, Newton’s method, conditional gradient and sub-gradient optimization, interior-point methods and penalty and barrier methods.

MATH 708 – Partial Differential Equations (3-0-3)

Many laws of physics are formulated as partial differential equations (PDEs). This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.

MATH 709 – Probability and Stochastic Processes (3-0-3)

This course aims at presenting the mathematically rigorous concepts underlying classical continuous-time stochastic processes. In the first part of the course, some fundamental notions of modern probability measure theory are revised together with the notion of measure-theoretic derivative. Then, fundamental topics of stochastic processes, such as martingales, Kolmogorov鈥檚 extension and Brownian motions, will be presented.

MATH 710 – Selected topics in group theory (3-0-3)

The course gives an up-to-date and modern overview of the main concepts in group theory. Group theoretical properties and several examples of groups arising in many branches of mathematics, physics, and chemistry are studied to show how group theory emerges in different mathematical fields and in applied sciences. Group theory and the closely related representation theory have many applications in physics and chemistry since various physical systems, such as crystals and the hydrogen atom, can be modeled by symmetry groups.

MATH 711 – Selected Topics in High Dimensional Statistics (3-0-3)

Technological and scientific advances in our ability to collect, observe, and store data throughout science, engineering, and commerce call for a change in the basic understanding of how we are to learn and handle data.聽 This course rigorously surveys the modern literature concerning the mathematical foundations of several statistical learning and inference problems. A particular emphasis is on non-asymptotic results. Topics covered include sparse recovery, high dimensional PCA, and nonparametric least squares. The aim is towards developing algorithms that are effective both in theory and in applications.

MATH 712 – Quantitative Principles in Biology (3-0-3)

This is an advanced course in the general field of Mathematical and Physical Biology and related topics. The course attempts to identify candidate quantitative principles for Biology. The mathematical tools used in this course span from stochastic processes, probability, network and information theory. The material presented here is beyond the classical courses of Mathematical Biology and meets the current research trends.

MATH 713 – Measure Theory (3-0-3)

Measure theory provides a foundation for many branches of mathematics such as harmonic analysis, ergodic theory, theory of partial differential equations and probability theory. It is a central, extremely useful part of modern analysis. This course is an introduction to abstract measure theory and the Lebesgue integral. The Lebesgue integral is introduced, and the main convergence theorems proved. Emphasis is given to the construction of the Lebesgue measure in Rn. Other topics treated in the course are Lp-spaces, the Radon-Nikodym theorem, the Lebesgue differentiation theorem, and the Fubini theorem.

MATH 714 – Real Analysis (3-0-3)

This course is at a higher level compared to MATH 324 and covers more advanced topics as well the standard topics more deeply. In this course, students will be introduced to the difference between pointwise and uniform convergence. The integration and differentiation of a sequence of functions will be treated in depth. The course provides a foundation on how to employ the Riemann-Stieltjes integral, and how to use the Weierstrass M-test for series of functions. Determining whether a series of functions converges uniformly will be discussed as well as the facility with power series of functions and their use in solving differential equations. This course also provides the students with the ability to demonstrate familiarity with common pathological counterexamples regarding sequences and series of functions. It聽 introduces the students to Riemann integration and its use in analysis, various convergence theorems and their applications, equicontinuity and the Arzela-Ascoli theorem, the Arzela-Ascoli theorem to ODEs, and how to use the Stone-Weierstrass theorem in practical situations.

MATH 715 – Analytical Foundations of Risk and Optimization (3-0-3)

The course provides a solid and rigorous introduction to analytical and optimization methods of primary importance in modern economic and financial applications. Recent evolution of the concept and taxonomy of risk provides a comprehensive mathematical framework to evaluate and assess the exposure to sources of risk whose impact may severely jeopardize effective decision making. The course additionally includes recently developed methods for optimal decision making under uncertainty.

MATH 716 – Differential Geometry and its Applications (3-0-3)

The course gives an up-to-date and modern overview of the main concepts of differential geometry and its applications to mechanics, geometric control and soft matter and soft robotics problems. The course also introduces elementary ideas of topological physics.

MATH 717 – Methods of Mathematical Physics (3-0-3)

This course will combine theoretical physics with high-level math courses in differential equations, vector calculus, and applied mathematics. The course will also get plenty of chances to apply that learning with hands-on labs in computer programming, optics, and provides an introduction to mathematical aspects of Quantum Mechanics and Quantum Field Theory, and to make some fundamental topics in this research area accessible to graduate students with interests in Analysis, Mathematical Physics, PDEs, and Applied Mathematics.

MATH 722 – Efficient Algorithms for Convex Programming (3-0-3)

This is an advanced course on efficient techniques for solving convex programming problems, which covers a range of topics at the intersection of Mathematical Programming and Algorithm Design. The focus is on how efficiently to solve linear/convex programs, with emphasis on computational complexity.

MATH 725 – Computational Systems Biology (3-0-3)

This course will introduce PhD students to computational and mathematical modelling of oncology. In particular, we will cover topics related to cancer phenomena at different spatiotemporal scales, such as cancer genetics and signaling pathways, cancer cell interactions and clinical tumor growth.

MATH 735 – Risk analysis, Stochastic Optimization and Reinforcement Learning (3-0-3)

The course provides a rigorous introduction to optimization methods and mathematical statistics relevant in the domain of applied optimization based on risk-reward decision rules, thus encompassing risk models in commodity and financial markets, and associated one or multi-period optimization problems. Recent evolution of the concept and taxonomy of risk provides a comprehensive mathematical framework to evaluate and assess the exposure to sources of risk whose impact may severely jeopardize effective decision making. The course includes recently developed methods for optimal decision making under uncertainty. Particular emphasis is given to decision models in a dynamic, multiperiod framework.

MATH 777 – Mathematical Models for Biology and Epidemiology (3-0-3)

This is an advanced course in Mathematical Modelling for the Biosciences with emphasis on Epidemiology and related topics. The instructor should select timely topics and adapt the course to the needs of the students from a vast portfolio of research areas including population dynamics, epidemiology, modelling of infectious diseases and model calibration and validation.

MATH 787 – Mathematical Imaging (3-0-3)

This course will cover the mathematical conceptual foundations of medical imaging science. These fundamentals cover topics such as signal processing, noise analysis, and image reconstruction with a large focus on Fourier Transforms. The course includes theoretical foundations and concept applications through exercises based on computational environment such as Matlab. The course includes also a project compiling all the fundamental concepts introduced for real data applications such as medical and engineering research challenges.